Magic constant

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The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general where is the side length of the square.

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Magicsquareexample.svg

For normal magic squares of orders n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8x8 square will always equate to 260 for each row, column, or diagonal.

The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes. Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give polyiamond magic constant. [1]

Magic stars

The magic constant of an n-pointed normal magic star is .

Magic series

In 2013 Dirk Kinnaes found the magic series polytope. The number of unique sequences that form the magic constant is now known up to . [2]

Physics

In the mass model, the value in each cell specifies the mass for that cell. [3] This model has two notable properties. First it demonstrates the balanced nature of all magic squares. If such a model is suspended from the central cell the structure balances. ( consider the magic sums of the rows/columns .. equal mass at an equal distance balance). The second property that can be calculated is the moment of inertia. Summing the individual moments of inertia ( distance squared from the center x the cell value) gives the moment of inertia for the magic square. "This is the only property of magic squares, aside from the line sums, which is solely dependent on the order of the square." [4]

See also

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In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the so-called magic constant of the cube, denoted M3(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant

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  2. All pairs of integers distant n/2 along any diagonal are complementary.

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Water retention on mathematical surfaces is the catching of water in ponds on a surface of cells of various heights on a regular array such as a square lattice, where water is rained down on every cell in the system. The boundaries of the system are open and allow water to flow out. Water will be trapped in ponds, and eventually all ponds will fill to their maximum height, with any additional water flowing over spillways and out the boundaries of the system. The problem is to find the amount of water trapped or retained for a given surface. This has been studied extensively for two mathematical surfaces: magic squares and random surfaces. The model can also be applied to the triangular grid.

(2,1)-Pascal triangle

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References

  1. http://oeis.org/A303295/
  2. Walter Trump http://www.trump.de/magic-squares/
  3. Heinz http://www.magic-squares.net/ms-models.htm#A 3 dimensional magic square/
  4. Peterson http://www.sciencenews.org/view/generic/id/7485/description/Magic_Square_Physics/